100% pixel perfect Displacement map
Did you make a UV map for your plain (plane?)?
If I understand correctly, your plane is square in the projection along the z-axis.
You could switch to top view, switch to edit mode, UV-unwrap with the option &quot;project from view (bounds)&quot;. This should line up everything neatly.
Maybe you'll have to rotate the UV map in the UV editor by some multiple of 90 degrees afterwards, but the size and relative position should match then.
This is how you set the texture coordinates to use the UVMap:

Or maybe upload your blendfile, so we can have a better look?